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...Cpscn randomized algorithms term notes on symmetric matrices prof nick harvey university of british columbia wereview some basic results concerning all that we discuss are over the real numbers let a be matrix size d and i denote identity perhaps most important useful property is their eigenvalues behave very nicely denition u called an orthogonal if utu this implies uut by uniqueness inverses fact spectral theorem any there exists diagonal such udut decomposition ui ith column entry then ud orthonormal basis consisting eigenvectors eigenvalue corresponding to can also write x t iuiu uniquely determined up reordering caution product two usually not positive semi denite its non negative denoted where here denotes zero strictly many equivalent characterizations deniteness one condition ax r similarly xtax rd another only v used dene partial ordering l owner or for b vtav vtbv vectors proof note holds vt whichis set closed under addition scaling convex cone scalars follows easily from doe...